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a) Problem consideration and coordinate systems for the original configuration and the rotated configuration (after Tran Manh et al. 2015) and b) Conformal mapping of points from the physical z-plane onto the unit circle exterior in the ζ -plane .
Source publication
The implementation of closed-form solutions for stress and displacement fields around tunnels with arbitrary geometry, often based on the complex variable theory and the method of conformal mapping, can be quite challenging from a mathematical point of view. In this paper a solution strategy for the implementation of a chosen closed-form solution f...
Contexts in source publication
Context 1
... the solution of Tran Manh et al. (2015) the initial problem consists of a tunnel with an arbitrary geometry (in our case semicircular) excavated in an infinite elastic and transversely isotropic rock mass. The planes of isotropy are oriented at an angle β from the horizontal x* -axis along the local x´-y´-coordinate system in the original configuration (see Figure 1a). The initial stresses are assumed to be homogeneous and anisotropic with a vertical stress σ0 and a horizontal stress K0⋅σ0. ...
Context 2
... rotated configuration (x-y-system), later referred to as the z-plane, is the configuration considered for the conformal mapping and the evaluation of the elastic potentials. In Fig 1b) the mapping process of points from the z-plane to the exterior of a unit circle on the ζ-plane is shown. In the solution of Tran Manh et al. (2015), the mapping of points p on the tunnel exterior in the z-plane, expressed in terms of the complex variable zp=xp+iyp, is carried out to the outside of a unit circle (p expressed as ζp=ρpe iθp with polar radius ρp and polar angle θp) based on the negative powers of the Laurent series (Eqn.1). ...
Context 3
... the derivatives of the stress potentials Ωk, are found for each point on the z-plane, the stress and displacement field changes arising from the relaxation of the internal pressure by a stress release factor λ can be determined using formulae presented in the paper of Tran Manh et al. (2015). To receive the final stress field for the given problem in the rotated configuration (see Figure 1a) the computed changes in the stresses must be superimposed on the initial values for the stress components. In a final step, the whole system, including the geometry and the stress and displacement fields must be rotated back into the original configuration. ...
Context 4
... 3950 Figure 4. Contour plots for resulting a) displacements and b) stresses from the numerical and analytical solutions in the x*-y*-coordinate system. Figure 4 compares the numerical and analytical results in terms of contour plots of displacements and stresses in the global Cartesian system (original configuration acc. to Figure 1). Overall, a very good agreement between the results from both calculation approaches can be seen. ...
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Citations
... Various available stress-displacement solutions for tunnels with arbitrary geometries in isotropic and anisotropic grounds rely on the complex variable theory and the method of conformal mapping. However, the implementation of these solutions is accompanied by practical difficulties, e.g. when mathematical optimization problems have to be solved (see Winkler et al. 2023). ...
In case of tunnels with arbitrary geometries, solutions for stresses and displacements in the tunnel exterior might be derived with the aid of the conformal mapping technique of the complex variable method. Thereby, the physical tunnel domain is mapped onto a fictitious unit circle domain on which the elastic potentials, as part of the final solution, are evaluated. The used mapping function involves complex mapping coefficients. In this paper an overview of analytical solutions for stress and displacements fields around tunnels is provided, from the early Kirsch solution to the solutions involving the complex variable theory and conformal mapping. A possible solution procedure for the determination of these mapping coefficients based on an iterative process including the solution of linear systems of equations is presented. The proposed solution procedure can be utilized for the determination of the mapping coefficients of various conformal mapping functions as defined in different closed-form solutions.
